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2013 | Buch

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Discrete–Time Stochastic Control and Dynamic Potential Games

The Euler–Equation Approach

verfasst von: David González-Sánchez, Onésimo Hernández-Lerma

Verlag: Springer International Publishing

Buchreihe : SpringerBriefs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

There are several techniques to study noncooperative dynamic games, such as dynamic programming and the maximum principle (also called the Lagrange method). It turns out, however, that one way to characterize dynamic potential games requires to analyze inverse optimal control problems, and it is here where the Euler equation approach comes in because it is particularly well–suited to solve inverse problems. Despite the importance of dynamic potential games, there is no systematic study about them. This monograph is the first attempt to provide a systematic, self–contained presentation of stochastic dynamic potential games.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction and Summary

Abstract

This monograph concerns discrete-time optimal control problems (OCPs) and stochastic games with an infinite horizon. One of our main objectives is to identify some stochastic games, called dynamic potential games, in which Nash equilibria can be found by solving a single OCP. In this chapter we introduce the basic concepts of OCPs and stochastic games by means of examples. We also provide an example of a potential game, namely, the stochastic lake game (SLG) of Dechert and O’Donnell [23]. Likewise, we present some related literature about solution methods for OCPs as well as some basic ideas about static potential games. We close the chapter by describing the contents of the remaining chapters.

David González-Sánchez, Onésimo Hernández-Lerma

Chapter 2. Direct Problem: The Euler Equation Approach

Abstract

This chapter concerns deterministic and stochastic nonstationary discrete-time optimal control problems (OCPs) with an infinite horizon. We show, using Gâteaux differentials, that the so-called Euler equation (EE) and a transversality condition (TC) are necessary conditions for optimality. In particular, the TC is obtained in a more general form and under milder hypotheses than in previous works. Sufficient conditions are also provided. We find closed-form solutions to several (discounted) stationary and nonstationary control problems. The results in this chapter come from González–Sánchez and Hernández–Lerma [37].

David González-Sánchez, Onésimo Hernández-Lerma

Chapter 3. The Inverse Optimal Control Problem

Abstract

In this chapter we study an inverse optimal control problem in discrete-time stochastic control. We give necessary and sufficient conditions for a solution to a system of stochastic difference equations to be the solution of a certain OCP. Our results extend to the stochastic case the work of Dechert [21]. In particular, we present a stochastic version of an important principle in welfare economics. The presentation of this chapter is based on González–Sánchez and Hernández–Lerma [36].

David González-Sánchez, Onésimo Hernández-Lerma

Chapter 4. Dynamic Games

Abstract

The purpose of this chapter is twofold. First, to extend the Euler equation (EE) approach, which was studied in Chaps. 2 and 3 for optimal control problems (OCPs), to find Nash equilibria in dynamic games. Second, to identify classes of dynamic potential games (DPGs), that is, games with Nash equilibria that can be found by solving a single OCP. In particular, the stochastic lake game (SLG) of Example 1.2 is included in one of these classes.

David González-Sánchez, Onésimo Hernández-Lerma

Chapter 5. Conclusions and Suggestions for Future Research

Abstract

In this book we have studied discrete-time stochastic optimal control problems (OCPs) and dynamic games by means of the Euler equation (EE) approach. Both direct and inverse problems in optimal control were considered in Chaps. 2 and 3, respectively. In Chap. 4 we dealt with dynamic games. Some of our main results are mentioned below in addition to discussing their relevance and possible generalizations.

David González-Sánchez, Onésimo Hernández-Lerma

Backmatter

Titel: Discrete–Time Stochastic Control and Dynamic Potential Games
verfasst von: David González-Sánchez
Onésimo Hernández-Lerma
Copyright-Jahr: 2013
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-01059-5
Print ISBN: 978-3-319-01058-8
DOI: https://doi.org/10.1007/978-3-319-01059-5